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Cyclotron Resonance and Energy Band Structures

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Basic Semiconductor Physics

Part of the book series: Graduate Texts in Physics ((GTP))

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Abstract

Cyclotron resonance experiments were successfully used to determine the effective masses of electrons in the conduction band and holes in the valence bands. Electrons are subject to rotational motion in a magnetic field (cyclotron motion) and resonantly absorb the radiation fields (microwaves or infrared radiation) when the cyclotron frequency is equal to the radiation frequency. The resonant condition gives the effective mass of the electron. Analyzing the cyclotron resonance data of Ge and Si, detailed information of the electrons in the conduction band valleys. In addition, the analysis of the hole masses based on the \(\varvec{k}\cdot \varvec{p}\) perturbation reveals the detailed valence band structures such as the heavy hole, light hole, and spin–orbit split–off bands. The non-parabolicity of the conduction band is also discussed. Quantum mechanical treatment of the electrons in the conduction band and holes in the valence bands leads us to draw the picture of Landau levels which is used to understand magnetotransport in Chap. 7 and quantum Hall effect in Chap. 8.

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Notes

  1. 1.

    See also Sect. 1.7 of Chap. 1 and (1.146) of p. 51.

  2. 2.

    In many references the average effective mass is expressed as

    $$\begin{aligned} \frac{m}{m^*} = -A \pm \sqrt{B^2 + \frac{C^2}{6}} \,, \end{aligned}$$

    which should be corrected as above.

  3. 3.

    Note that P used in Tables 1.9 and 1.11 is \(P_\mathrm{au} = 2\langle \varGamma _{2^\prime }\left| p_x \right| X \rangle =2P_{0}\) and thus \({\mathcal E}_{P0}=(2/m)P_{0}^2=P_\mathrm{au}^2/2\) m  which is equivalent to \(P_\mathrm{au}^2\) in atomic units.

  4. 4.

    In Sect. 2.6 we deal with the Luttinger parameters and introduce \({\mathcal E}_{P0}=(2/m)P_{0}^2\) with the momentum matrix element \(P_0 = \langle \varGamma _{2'}\vert p_x \vert X\rangle \). The parameter \({\mathcal E}_{P0}\) is often cited for the analysis of the valence band states and of the Luttinger parameters [3, 6]. The values of \({\mathcal E}_{P0}\) for various semiconductors range from 19 to 27 eV. The matrix element introduced by Cardona and Pollak [5] is \(P=2P_{0}\) and \({\mathcal E}_{P0}= P^2/2m\). Also note the difference between the subscripts of \({\mathcal {E}}_p\) and of \({\mathcal {E}}_{P0}\).

  5. 5.

    The matrix elements derived by using the expressions of Dresselhaus et al. are equivalent to the expressions of Luttinger and Kohn except the diagonal elements as discussed before, and the corresponding secular equation of Luttinger and Kohn Hamiltonian gives eigenvalue \({\mathcal {E}}(\varvec{k})\), while the secular equation of Dresselhaus, Kip and Kittel gives \(\lambda ={\mathcal {E}}(\varvec{k})-\hbar ^2k^2/2m\).

  6. 6.

    The reason why the upper four conduction bands appear in the matrix elements is understood from the the selection rules \(\langle \varGamma _{25'}\vert \varvec{p}\vert \varGamma _{l,j}\rangle \), where \(\varvec{p}\) is momentum operator and transforms as the representation \(\varGamma _{15}\), and \(\varGamma _{l,j}\) are upper conduction band states. The direct product is given by using the character table of Table 1.4

    $$ \varGamma _{25'} \times \varGamma _{15} = \varGamma _{12'}+\varGamma _{15}+\varGamma _{2'}+\varGamma _{25} \,, $$

    and thus only the conduction band states of the four representations on the right hand perturb the valence band edge.

  7. 7.

    Equations (2.179a) \(\sim \) (2.179c) are obtained by neglecting the spin–orbit splitting, or by putting \(\varDelta _0=0\) in (2.234a) \(\sim \) (2.234c).

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  1. Osaka University, Suita, Osaka, Japan

    Chihiro Hamaguchi

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Hamaguchi, C. (2017). Cyclotron Resonance and Energy Band Structures. In: Basic Semiconductor Physics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-66860-4_2

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